We use two non-Riemannian curvature tensors, the χ-curvature and mean Berwald to characterise a class of Finsler metrics admitting first integrals. This includes constant flag curvature.

In this paper, we study the class of cubic (\alpha, \beta)-metrics. We show that every weakly Landsberg \beta)-metric has vanishing S-curvature. Using it, prove is a metric if and only it Berwald metric. This yields an extension Matsumoto's result for

We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler-Lagrange equation for this functional is an elliptic equation for Hermitian metrics. Solutions to this equation are related to Kähler-Einstein metrics, and are automatically Kähler-Einstein under certain conditions. Given this, a natural parabolic flow equation arises. We prove short time existen...